A324438 a(n) = Product_{i=1..n, j=1..n} (i^5 + j^5).

1, 2, 139392, 305013568273920000, 1174837791623127613548781790822400000000, 139642003782073074626249921818187528362524804267528306032640000000000000

Offset: 0

Author

Vaclav Kotesovec, Feb 28 2019

Keywords

• nonn

Crossrefs

Cf. A079478, A324403, A324426, A324437, A324439, A324440, A367834.

Cf. A272248, A367679.

Programs

Maple

a:= n-> mul(mul(i^5 + j^5, i=1..n), j=1..n):
seq(a(n), n=0..5);  # Alois P. Heinz, Nov 26 2023

Mathematica

Table[Product[i^5 + j^5, {i, 1, n}, {j, 1, n}], {n, 1, 6}]

Python

from math import prod, factorial
def A324438(n): return prod(i**5+j**5 for i in range(1,n) for j in range(i+1,n+1))**2*factorial(n)**5<Chai Wah Wu, Nov 26 2023

Formula

a(n) ~ c * 2^(2*n*(n+1)) * phi^(sqrt(5)*n*(n+1)) * exp(Pi*sqrt(phi)*n*(n+1)/5^(1/4) - 15*n^2/2) * n^(5*n^2 - 5/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio and c = 0.1574073828647726237455544898360432469056972905505624900871695...

a(n) = A367679(n) * A079478(n). - Vaclav Kotesovec, Nov 26 2023

For n>0, a(n)/a(n-1) = A272248(n)^2 / (2*n^15). - Vaclav Kotesovec, Dec 02 2023

Extensions

a(0)=1 prepended by Alois P. Heinz, Nov 26 2023